Proem

One of the many intriguing observations in Miranda Lund's book Sacred Geometry is that there are only three regular tilings of a plane. That is, triangles, squares, and hexagons are the only three regular polygons that can completely fill a plane if repeated over and over.

There are other ways of filling a plane with combinations of two or more different types of regular polygons, known as semi-regular and demi-regular tilings. But these semi- and demi-regular tilings involve only triangles, squares, hexagons, octagons, or dodecagons: that is, 3, 4, 6, 8, and 12 sided figures. What became of the 5, 7, 9, 10, and 11 sided figures? Assuming that a desperate soul were to try to fill a plane with these figures as regularly as possible, what is the best they could do?

It turns out there are a good many visually interesting arrangements of polygons across a plane that are structurally interesting, without being a regular, demi-regular, or semi-regular tiling. Here's an example that is based on pentagons:

Some of the interesting properties of this arrangement are:

  • All the line segments are of equal length.
     
  • All line segments meet at their points; no line segment connects to the middle of another line segment.
     
  • The entire arrangement arises from placing pentagons so they touch each other at their points. No extra lines are drawn. Every line segment is the side of a pentagon.
     
  • There are only four shapes in the pattern: regular polygon, regular triangle, parallelogram, and a six-pointed star whose points are a bit sharper than in a regular hexagram. Of these, the triangles, parallelogram, and stars are formed from the negative space between the pentagons.

I don't know of a mathematical term for tilings with these types of properties, but I have discovered (or possibly rediscovered) many of them by placing regular polygons point to point. For a time, it became an obsession of sorts to enumerate these various patterns. Polygon Madness is the record of my experiments with such figures.

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